In mathematics and computer science, an algorithm is an unambiguous specification of how to solve a class of problems. Algorithms can perform calculation, data processing, automated reasoning, other tasks; as an effective method, an algorithm can be expressed within a finite amount of space and time and in a well-defined formal language for calculating a function. Starting from an initial state and initial input, the instructions describe a computation that, when executed, proceeds through a finite number of well-defined successive states producing "output" and terminating at a final ending state; the transition from one state to the next is not deterministic. The concept of algorithm has existed for centuries. Greek mathematicians used algorithms in the sieve of Eratosthenes for finding prime numbers, the Euclidean algorithm for finding the greatest common divisor of two numbers; the word algorithm itself is derived from the 9th century mathematician Muḥammad ibn Mūsā al-Khwārizmī, Latinized Algoritmi.
A partial formalization of what would become the modern concept of algorithm began with attempts to solve the Entscheidungsproblem posed by David Hilbert in 1928. Formalizations were framed as attempts to define "effective calculability" or "effective method"; those formalizations included the Gödel–Herbrand–Kleene recursive functions of 1930, 1934 and 1935, Alonzo Church's lambda calculus of 1936, Emil Post's Formulation 1 of 1936, Alan Turing's Turing machines of 1936–37 and 1939. The word'algorithm' has its roots in Latinizing the name of Muhammad ibn Musa al-Khwarizmi in a first step to algorismus. Al-Khwārizmī was a Persian mathematician, astronomer and scholar in the House of Wisdom in Baghdad, whose name means'the native of Khwarazm', a region, part of Greater Iran and is now in Uzbekistan. About 825, al-Khwarizmi wrote an Arabic language treatise on the Hindu–Arabic numeral system, translated into Latin during the 12th century under the title Algoritmi de numero Indorum; this title means "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name.
Al-Khwarizmi was the most read mathematician in Europe in the late Middle Ages through another of his books, the Algebra. In late medieval Latin, English'algorism', the corruption of his name meant the "decimal number system". In the 15th century, under the influence of the Greek word ἀριθμός'number', the Latin word was altered to algorithmus, the corresponding English term'algorithm' is first attested in the 17th century. In English, it was first used in about 1230 and by Chaucer in 1391. English adopted the French term, but it wasn't until the late 19th century that "algorithm" took on the meaning that it has in modern English. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu, it begins thus: Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. Which translates as: Algorism is the art by which at present we use those Indian figures, which number two times five; the poem is a few hundred lines long and summarizes the art of calculating with the new style of Indian dice, or Talibus Indorum, or Hindu numerals.
An informal definition could be "a set of rules that defines a sequence of operations". Which would include all computer programs, including programs that do not perform numeric calculations. A program is only an algorithm if it stops eventually. A prototypical example of an algorithm is the Euclidean algorithm to determine the maximum common divisor of two integers. Boolos, Jeffrey & 1974, 1999 offer an informal meaning of the word in the following quotation: No human being can write fast enough, or long enough, or small enough† to list all members of an enumerably infinite set by writing out their names, one after another, in some notation, but humans can do something useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human, capable of carrying out only elementary operations on symbols.
An "enumerably infinite set" is one whose elements can be put into one-to-one correspondence with the integers. Thus and Jeffrey are saying that an algorithm implies instructions for a process that "creates" output integers from an arbitrary "input" integer or integers that, in theory, can be arbitrarily large, thus an algorithm can be an algebraic equation such as y = m + n – two arbitrary "input variables" m and n that produce an output y. But various authors' attempts to define the notion indicate that the word implies much more than this, something on the order of: Precise instructions for a fast, efficient, "good" process that specifies the "moves" of "the computer" to find and process arbitrary input integers/symbols m and n, symbols + and =... and "effectively" produce, in a "reasonable" time, output-integer y at a specified place and in a specified format
In physics, in particular as measured by radiometry, radiant energy is the energy of electromagnetic and gravitational radiation. As energy, its SI unit is the joule; the quantity of radiant energy may be calculated by integrating radiant flux with respect to time. The symbol Qe is used throughout literature to denote radiant energy. In branches of physics other than radiometry, electromagnetic energy is referred to using E or W; the term is used when electromagnetic radiation is emitted by a source into the surrounding environment. This radiation may be invisible to the human eye; the term "radiant energy" is most used in the fields of radiometry, solar energy and lighting, but is sometimes used in other fields. In modern applications involving transmission of power from one location to another, "radiant energy" is sometimes used to refer to the electromagnetic waves themselves, rather than their energy. In the past, the term "electro-radiant energy" has been used; the term "radiant energy" applies to gravitational radiation.
For example, the first gravitational waves observed were produced by a black hole collision that emitted about 5.3×1047 joules of gravitational-wave energy. Because electromagnetic radiation can be conceptualized as a stream of photons, radiant energy can be viewed as photon energy – the energy carried by these photons. Alternatively, EM radiation can be viewed as an electromagnetic wave, which carries energy in its oscillating electric and magnetic fields; these two views are equivalent and are reconciled to one another in quantum field theory. EM radiation can have various frequencies; the bands of frequency present in a given EM signal may be defined, as is seen in atomic spectra, or may be broad, as in blackbody radiation. In the photon picture, the energy carried by each photon is proportional to its frequency. In the wave picture, the energy of a monochromatic wave is proportional to its intensity; this implies that if two EM waves have the same intensity, but different frequencies, the one with the higher frequency "contains" fewer photons, since each photon is more energetic.
When EM waves are absorbed by an object, the energy of the waves is converted to heat. This is a familiar effect, since sunlight warms surfaces that it irradiates; this phenomenon is associated with infrared radiation, but any kind of electromagnetic radiation will warm an object that absorbs it. EM waves can be reflected or scattered, in which case their energy is redirected or redistributed as well. Radiant energy is one of the mechanisms by which energy can leave an open system; such a system can be man-made, such as a solar energy collector, or natural, such as the Earth's atmosphere. In geophysics, most atmospheric gases, including the greenhouse gases, allow the Sun's short-wavelength radiant energy to pass through to the Earth's surface, heating the ground and oceans; the absorbed solar energy is re-emitted as longer wavelength radiation, some of, absorbed by the atmospheric greenhouse gases. Radiant energy is produced in the sun as a result of nuclear fusion. Radiant energy is used for radiant heating.
It can be generated electrically by infrared lamps, or can be absorbed from sunlight and used to heat water. The heat energy is emitted from a warm element and warms people and other objects in rooms rather than directly heating the air; because of this, the air temperature may be lower than in a conventionally heated building though the room appears just as comfortable. Various other applications of radiant energy have been devised; these include treatment and inspection and sorting, medium of control, medium of communication. Many of these applications involve a source of radiant energy and a detector that responds to that radiation and provides a signal representing some characteristic of the radiation. Radiant energy detectors produce responses to incident radiant energy either as an increase or decrease in electric potential or current flow or some other perceivable change, such as exposure of photographic film
An estuary is a enclosed coastal body of brackish water with one or more rivers or streams flowing into it, with a free connection to the open sea. Estuaries form a transition zone between river environments and maritime environments, they are subject both to marine influences—such as tides and the influx of saline water—and to riverine influences—such as flows of fresh water and sediment. The mixing of sea water and fresh water provide high levels of nutrients both in the water column and in sediment, making estuaries among the most productive natural habitats in the world. Most existing estuaries formed during the Holocene epoch with the flooding of river-eroded or glacially scoured valleys when the sea level began to rise about 10,000–12,000 years ago. Estuaries are classified according to their geomorphological features or to water-circulation patterns, they can have many different names, such as bays, lagoons, inlets, or sounds, although some of these water bodies do not meet the above definition of an estuary and may be saline.
The banks of many estuaries are amongst the most populated areas of the world, with about 60% of the world's population living along estuaries and the coast. As a result, many estuaries suffer degradation from a variety of factors including: sedimentation from soil erosion from deforestation and other poor farming practices; the word "estuary" is derived from the Latin word aestuarium meaning tidal inlet of the sea, which in itself is derived from the term aestus, meaning tide. There have been many definitions proposed to describe an estuary; the most accepted definition is: "a semi-enclosed coastal body of water, which has a free connection with the open sea, within which sea water is measurably diluted with freshwater derived from land drainage". However, this definition excludes a number of coastal water bodies such as coastal lagoons and brackish seas. A more comprehensive definition of an estuary is "a semi-enclosed body of water connected to the sea as far as the tidal limit or the salt intrusion limit and receiving freshwater runoff.
This broad definition includes fjords, river mouths, tidal creeks. An estuary is a dynamic ecosystem having a connection to the open sea through which the sea water enters with the rhythm of the tides; the sea water entering the estuary streams. The pattern of dilution varies between different estuaries and depends on the volume of fresh water, the tidal range, the extent of evaporation of the water in the estuary. Drowned river valleys are known as coastal plain estuaries. In places where the sea level is rising relative to the land, sea water progressively penetrates into river valleys and the topography of the estuary remains similar to that of a river valley; this is the most common type of estuary in temperate climates. Well-studied estuaries include the Severn Estuary in the United Kingdom and the Ems Dollard along the Dutch-German border; the width-to-depth ratio of these estuaries is large, appearing wedge-shaped in the inner part and broadening and deepening seaward. Water depths exceed 30 m.
Examples of this type of estuary in the U. S. are the Hudson River, Chesapeake Bay, Delaware Bay along the Mid-Atlantic coast, Galveston Bay and Tampa Bay along the Gulf Coast. Bar-built estuaries are found in place where the deposition of sediment has kept pace with rising sea level so that the estuaries are shallow and separated from the sea by sand spits or barrier islands, they are common in tropical and subtropical locations. These estuaries are semi-isolated from ocean waters by barrier beaches. Formation of barrier beaches encloses the estuary, with only narrow inlets allowing contact with the ocean waters. Bar-built estuaries develop on sloping plains located along tectonically stable edges of continents and marginal sea coasts, they are extensive along the Atlantic and Gulf coasts of the U. S. in areas with active coastal deposition of sediments and where tidal ranges are less than 4 m. The barrier beaches that enclose bar-built estuaries have been developed in several ways: building up of offshore bars by wave action, in which sand from the sea floor is deposited in elongated bars parallel to the shoreline, reworking of sediment discharge from rivers by wave and wind action into beaches, overwash flats, dunes, engulfment of mainland beach ridges due to sea level rise and resulting in the breaching of the ridges and flooding of the coastal lowlands, forming shallow lagoons, elongation of barrier spits from the erosion of headlands due to the action of longshore currents, with the spits growing in the direction of the littoral drift.
Barrier beaches form in shallow water and are parallel to the shoreline, resulting in long, narrow estuaries. The average water depth is less than 5 m, exceeds 10 m. Examples of bar-built estuaries are Barnegat Bay, New Jersey. Fjords were formed where pleistocene glaciers deepened and widened existing river valleys so that they become U-shaped in cross s
Silt is granular material of a size between sand and clay, whose mineral origin is quartz and feldspar. Silt may occur as a soil or as sediment mixed in suspension with water and soil in a body of water such as a river, it may exist as soil deposited at the bottom of a water body, like mudflows from landslides. Silt has a moderate specific area with a non-sticky, plastic feel. Silt has a floury feel when dry, a slippery feel when wet. Silt can be visually observed with a hand lens, it can be felt by the tongue as granular when placed on the front teeth. Silt is created by a variety of physical processes capable of splitting the sand-sized quartz crystals of primary rocks by exploiting deficiencies in their lattice; these involve chemical weathering of rock and regolith, a number of physical weathering processes such as frost shattering and haloclasty. The main process is abrasion through transport, including fluvial comminution, aeolian attrition and glacial grinding, it is in semi-arid environments.
Silt is sometimes known as "rock flour" or "stone dust" when produced by glacial action. Mineralogically, silt is composed of quartz and feldspar. Sedimentary rock composed of silt is known as siltstone. Liquefaction created by a strong earthquake is silt suspended in water, hydrodynamically forced up from below ground level. In the Udden–Wentworth scale, silt particles range between 0.0039 and 0.0625 mm, larger than clay but smaller than sand particles. ISO 14688 grades silts between 0.063 mm. In actuality, silt is chemically distinct from clay, unlike clay, grains of silt are the same size in all dimensions. Clays are formed from thin plate-shaped particles held together by electrostatic forces, so present a cohesion. Pure silts are not cohesive. According to the U. S. Department of Agriculture Soil Texture Classification system, the sand–silt distinction is made at the 0.05 mm particle size. The USDA system has been adopted by the Agriculture Organization. In the Unified Soil Classification System and the AASHTO Soil Classification system, the sand–silt distinction is made at the 0.075 mm particle size.
Silts and clays are distinguished mechanically by their plasticity. Silt is transported in water or other liquid and is fine enough to be carried long distances by air in the form of dust. Thick deposits of silty material resulting from deposition by aeolian processes are called loess. Silt and clay contribute to turbidity in water. Silt is transported by water currents in the ocean; when silt appears as a pollutant in water the phenomenon is known as siltation. Silt, deposited by annual floods along the Nile River, created the rich, fertile soil that sustained the Ancient Egyptian civilization. Silt deposited by the Mississippi River throughout the 20th century has decreased due to a system of levees, contributing to the disappearance of protective wetlands and barrier islands in the delta region surrounding New Orleans. In southeast Bangladesh, in the Noakhali district, cross dams were built in the 1960s whereby silt started forming new land called "chars"; the district of Noakhali has gained more than 73 square kilometres of land in the past 50 years.
With Dutch funding, the Bangladeshi government began to help develop older chars in the late 1970s, the effort has since become a multi-agency operation building roads, embankments, cyclone shelters and ponds, as well as distributing land to settlers. By fall 2010, the program will have allotted some 100 square kilometres to 21,000 families. A main source of silt in urban rivers is disturbance of soil by construction activity. A main source in rural rivers is erosion from plowing of farm fields, clearcutting or slash and burn treatment of forests; the fertile black silt of the Nile river's banks is a symbol of rebirth, associated with the Egyptian god Anubis. Erosion control Nonpoint source pollution Sediment control Silt fence Siltation
The Cerrado is a vast tropical savanna ecoregion of Brazil in the states of Goiás, Mato Grosso do Sul, Mato Grosso and Minas Gerais. The Cerrado biome core areas are the plateaus in the center of Brazil; the main habitat types of the Cerrado include: forest savanna, wooded savanna, park savanna and gramineous-woody savanna. Savanna wetlands and gallery forests are included; the second largest of Brazil's major habitat types, after the Amazonian rainforest, the Cerrado accounts for a full 21 percent of the country's land area. The first detailed account of the Brazilian cerrados was provided by Danish botanist Eugenius Warming in the book Lagoa Santa, in which he describes the main features of the cerrado vegetation in the state of Minas Gerais. Since vast amounts of research have proved that the Cerrado is one of the richest of all tropical savanna regions and has high levels of endemism. Characterized by enormous ranges of plant and animal biodiversity, World Wide Fund for Nature named it the biologically richest savanna in the world, with about 10,000 plant species and 10 endemic bird species.
There are nearly 200 species of mammal in the Cerrado. The Cerrado's climate is typical of the wetter savanna regions of the world, with a semi-humid tropical climate; the Cerrado is limited to two dominant seasons throughout the year and dry. Annual temperatures for the Cerrado average between 22 and 27 °C and average precipitation between 800–2000 mm for over 90% of the area; this ecoregion has a strong dry season during the southern winter. The Cerrado is characterized by unique vegetation types, it is composed of a shifting mosaic of habitats, with the savanna-like cerrado itself on well-drained areas between strips of gallery forest which occur along streams. Between the cerrado and the gallery forest is an area of vegetation known as the wet campo with distinct up- and downslope borders where tree growth is inhibited due to wide seasonal fluctuations in the water table; the savanna portion of the Cerrado is heterogeneous in terms of canopy cover. Goodland divided the Cerrado into four categories ranging from least to most canopy cover: campo sujo, campo cerrado, cerrado sensu stricto and cerradao.
Around 800 species of trees are found in the Cerrado. Among the most diverse families of trees in the Cerrado are the Leguminosae, Myrtaceae and Rubiaceae. Much of the Cerrado is dominated by the Vochysiaceae due to the abundance of three species in the genus Qualea; the herbaceous layer reaches about 60 cm in height and is composed of the Poaceae, Leguminosae, Compositae and Rubiaceae. Much of the vegetation in the gallery forests is similar to nearby rainforest. Soil fertility, fire regime and hydrology are thought to be most influential in determining Cerrado vegetation. Cerrado soils are always well-drained and most are oxisols with low pH and low calcium and magnesium; the amount of potassium and phosphorus has been found to be positively correlated with tree trunk basal area in Cerrado habitats. Much as in other grasslands and savannas, fire is important in maintaining and shaping the Cerrado's landscape. Cerrado vegetation is believed to be ancient, stretching back as far in a prototypic form during the Cretaceous before Africa and South America separated.
A dynamic expansion and contraction between cerrado and Amazonian rainforest has occurred with expansion of the Cerrado during glacial periods like the Pleistocene. These processes and the resulting fragmentation have contributed to the high species richness both of the Cerrado and of the Amazonian rainforest; the insects of the Cerrado are understudied. A yearlong survey of the Cerrado at one reserve in Brazil found that the orders Coleoptera, Hymenoptera and Isoptera accounted for 89.5% of all captures. The Cerrado supports high density of leaf cutter ant nests which are very diverse. Along with termites, leaf cutter ants are the primary herbivores of the Cerrado and play an important role in consuming and decomposing organic matter, as well as constituting an important food source to many other animal species; the highest diversity of galling insects in the world is found in the Cerrado, with the most species found at the base of the Serro do Cipó in southeast Brazil. The Cerrado has a high diversity of vertebrates.
Lizard diversity is thought to be low in the Cerrado compared to other areas like caatinga or lowland rainforest although one recent study found 57 species in one cerrado area with the high diversity driven by the availability of open habitat. Ameiva ameiva is among the largest lizards found in the Cerrado and is the most important lizard predator where it is found in the Cerrado. There is a high diversity of snakes in the Cerrado with Colubridae being the richest family; the open nature of the cerrado vegetation most contrib
In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers. Turbulence is observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, most fluid flows occurring in nature or created in engineering applications are turbulent. Turbulence is caused by excessive kinetic energy in parts of a fluid flow, which overcomes the damping effect of the fluid's viscosity. For this reason turbulence is realized in low viscosity fluids. In general terms, in turbulent flow, unsteady vortices appear of many sizes which interact with each other drag due to friction effects increases; this increases the energy needed to pump fluid through a pipe. Turbulence can be exploited, for example, by devices such as aerodynamic spoilers on aircraft that "spoil" the laminar flow to increase drag and reduce lift.
The onset of turbulence can be predicted by the dimensionless Reynolds number, the ratio of kinetic energy to viscous damping in a fluid flow. However, turbulence has long resisted detailed physical analysis, the interactions within turbulence create a complex phenomenon. Richard Feynman has described turbulence as the most important unsolved problem in classical physics. Smoke rising from a cigarette. For the first few centimeters, the smoke is laminar; the smoke plume becomes turbulent as its Reynolds number increases with increases in flow velocity and characteristic lengthscale. Flow over a golf ball. If the golf ball were smooth, the boundary layer flow over the front of the sphere would be laminar at typical conditions. However, the boundary layer would separate early, as the pressure gradient switched from favorable to unfavorable, creating a large region of low pressure behind the ball that creates high form drag. To prevent this, the surface is dimpled to promote turbulence; this results in higher skin friction, but it moves the point of boundary layer separation further along, resulting in lower drag.
Clear-air turbulence experienced during airplane flight, as well as poor astronomical seeing. Most of the terrestrial atmospheric circulation; the oceanic and atmospheric mixed intense oceanic currents. The flow conditions in many industrial equipment and machines; the external flow over all kinds of vehicles such as cars, airplanes and submarines. The motions of matter in stellar atmospheres. A jet exhausting from a nozzle into a quiescent fluid; as the flow emerges into this external fluid, shear layers originating at the lips of the nozzle are created. These layers separate the fast moving jet from the external fluid, at a certain critical Reynolds number they become unstable and break down to turbulence. Biologically generated. Snow fences work by inducing turbulence in the wind, forcing it to drop much of its snow load near the fence. Bridge supports in water. In the late summer and fall, when river flow is slow, water flows smoothly around the support legs. In the spring, when the flow is faster, a higher Reynolds number is associated with the flow.
The flow may start off laminar but is separated from the leg and becomes turbulent. In many geophysical flows, the flow turbulence is dominated by the coherent structures and turbulent events. A turbulent event is a series of turbulent fluctuations that contain more energy than the average flow turbulence; the turbulent events are associated with coherent flow structures such as eddies and turbulent bursting, they play a critical role in terms of sediment scour and transport in rivers as well as contaminant mixing and dispersion in rivers and estuaries, in the atmosphere. In the medical field of cardiology, a stethoscope is used to detect heart sounds and bruits, which are due to turbulent blood flow. In normal individuals, heart sounds are a product of turbulent flow as heart valves close. However, in some conditions turbulent flow can be audible due to other reasons, some of them pathological. For example, in advanced atherosclerosis, bruits can be heard in some vessels that have been narrowed by the disease process.
Turbulence in porous media became a debated subject. Turbulence is characterized by the following features: Irregularity Turbulent flows are always irregular. For this reason, turbulence problems are treated statistically rather than deterministically. Turbulent flow is chaotic. However, not all chaotic flows are turbulent. Diffusivity The available supply of energy in turbulent flows tends to accelerate the homogenization of fluid mixtures; the characteristic, responsible for the enhanced mixing and increased rates of mass and energy transports in a flow is called "diffusivity". Turbulent diffusion is described by a turbulent diffusion coefficient; this turbulent diffusion coefficient is defined in a phenomenological sense, by analogy with the molecular diffusivities, but it does not have a true physical meaning, being dependent on the flow conditions, not a property of the fluid itself. In addition, the turbulent diffusivity concept assumes a con
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is thus the inverse of the spatial frequency. Wavelength is determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings and is a characteristic of both traveling waves and standing waves, as well as other spatial wave patterns. Wavelength is designated by the Greek letter lambda; the term wavelength is sometimes applied to modulated waves, to the sinusoidal envelopes of modulated waves or waves formed by interference of several sinusoids. Assuming a sinusoidal wave moving at a fixed wave speed, wavelength is inversely proportional to frequency of the wave: waves with higher frequencies have shorter wavelengths, lower frequencies have longer wavelengths. Wavelength depends on the medium. Examples of wave-like phenomena are sound waves, water waves and periodic electrical signals in a conductor.
A sound wave is a variation in air pressure, while in light and other electromagnetic radiation the strength of the electric and the magnetic field vary. Water waves are variations in the height of a body of water. In a crystal lattice vibration, atomic positions vary. Wavelength is a measure of the distance between repetitions of a shape feature such as peaks, valleys, or zero-crossings, not a measure of how far any given particle moves. For example, in sinusoidal waves over deep water a particle near the water's surface moves in a circle of the same diameter as the wave height, unrelated to wavelength; the range of wavelengths or frequencies for wave phenomena is called a spectrum. The name originated with the visible light spectrum but now can be applied to the entire electromagnetic spectrum as well as to a sound spectrum or vibration spectrum. In linear media, any wave pattern can be described in terms of the independent propagation of sinusoidal components; the wavelength λ of a sinusoidal waveform traveling at constant speed v is given by λ = v f, where v is called the phase speed of the wave and f is the wave's frequency.
In a dispersive medium, the phase speed itself depends upon the frequency of the wave, making the relationship between wavelength and frequency nonlinear. In the case of electromagnetic radiation—such as light—in free space, the phase speed is the speed of light, about 3×108 m/s, thus the wavelength of a 100 MHz electromagnetic wave is about: 3×108 m/s divided by 108 Hz = 3 metres. The wavelength of visible light ranges from deep red 700 nm, to violet 400 nm. For sound waves in air, the speed of sound is 343 m/s; the wavelengths of sound frequencies audible to the human ear are thus between 17 m and 17 mm, respectively. Note that the wavelengths in audible sound are much longer than those in visible light. A standing wave is an undulatory motion. A sinusoidal standing wave includes stationary points of no motion, called nodes, the wavelength is twice the distance between nodes; the upper figure shows three standing waves in a box. The walls of the box are considered to require the wave to have nodes at the walls of the box determining which wavelengths are allowed.
For example, for an electromagnetic wave, if the box has ideal metal walls, the condition for nodes at the walls results because the metal walls cannot support a tangential electric field, forcing the wave to have zero amplitude at the wall. The stationary wave can be viewed as the sum of two traveling sinusoidal waves of oppositely directed velocities. Wavelength and wave velocity are related just as for a traveling wave. For example, the speed of light can be determined from observation of standing waves in a metal box containing an ideal vacuum. Traveling sinusoidal waves are represented mathematically in terms of their velocity v, frequency f and wavelength λ as: y = A cos = A cos where y is the value of the wave at any position x and time t, A is the amplitude of the wave, they are commonly expressed in terms of wavenumber k and angular frequency ω as: y = A cos = A cos in which wavelength and wavenumber are related to velocity and frequency as: k = 2 π λ = 2 π f v = ω